# Linear Genomic Selection Indices

## Abstract

The linear genomic selection index (LGSI) is a linear combination of genomic estimated breeding values (GEBVs) used to predict the individual net genetic merit and select individual candidates from a non-phenotyped testing population as parents of the next selection cycle. In the LGSI, phenotypic and marker data from the training population are fitted into a statistical model to estimate all individual genome marker effects; these estimates can then be used in subsequent selection cycles to obtain GEBVs that are predictors of breeding values in a testing population for which there is only marker information.

Applying the LGSI requires predicting and ranking the net genetic merit of candidates for selection using only marker information. The combined LGSI (CLGSI), conversely, uses both phenotypic and GEBV information jointly to predict the net genetic merit. The CLGSI is typically used in the training populations to estimate true merit.

This vignette covers calculating LGSI and CLGSI using the `selection.index` package.

## 5.1 The Linear Genomic Selection Index

### Basic Conditions for Constructing the LGSI

To construct a valid LGSI, four essential conditions must be met: 1. All marker effects must be estimated simultaneously in the training population. 2. The estimated marker effects must be used in subsequent selection cycles to obtain GEBVs that act as predictors of the individual breeding values in the testing population (where there is only marker information). 3. The GEBV values must be composed entirely of the additive genetic effects. 4. Phenotypes must be used to estimate all marker effects in the training population, and not to make selections in the testing population (Heffner et al., 2009; Lorenz et al., 2011).

### The LGSI and Its Parameters

The objective of LGSI is to predict the net genetic merit \(H = \mathbf{w}'\mathbf{g}\), where \(\mathbf{g}\) is a vector of unobservable true breeding values, and \(\mathbf{w}\) is a vector of economic weights. Let \(\mathbf{\gamma}\) be the vector of known Genomic Estimated Breeding Values (GEBVs). The basic LGSI equation is:

\[ {I}_{\mathrm{G}}={\boldsymbol{\beta}}^{\prime}\boldsymbol{\gamma} \]

where \(\boldsymbol{\beta}\) is the unknown variable vector of optimal weights. For LGSI, it has been shown that maximizing the accuracy of \(\rho_{HI_G}\) results simply in \(\boldsymbol{\beta} = \mathbf{w}\).

The main advantage of the LGSI over the conventional Linear Phenotypic Selection Index (LPSI) lies in the possibility of reducing the intervals between selection cycles from roughly 4 years to 1.5 years in plant breeding. The selection response equation adjusts for cycle length \(L_G\):

\[ {R}_{I_G}=\frac{k_I}{L_G}\sqrt{{\mathbf{w}}^{\prime}\boldsymbol{\Gamma} \mathbf{w}} \] Where \(\boldsymbol{\Gamma}\) is the covariance matrix of the GEBVs.

### Statistical LGSI Properties

The LGSI operates as a direct genomic extension of the LPSI theory, retaining the exact same statistical properties: 1. The variance of \(I_G\) (\(\sigma_{I_G}^2\)) and the covariance between \(H\) and \(I_G\) (\(\sigma_{HI_G}\)) are equal (\(\sigma_{I_G}^2 = \sigma_{HI_G}\)). 2. The maximized correlation between \(H\) and \(I_G\) (the LGSI accuracy) is equal to \(\rho_{HI_G} = \frac{\sigma_{I_G}}{\sigma_H}\). 3. The variance of the predicted error, \(Var(H - I_G) = (1 - \rho_{HI_G}^2)\sigma_H^2\), is minimal. 4. The total variance of \(H\) explained by \(I_G\) is \(\sigma_{I_G}^2 = \rho_{HI_G}^2 \sigma_H^2\).

### LGSI vs LPSI Efficiency

To formalize the efficiency between the Genomic (LGSI) and Phenotypic (LPSI) selection index efficiency, we use the parameter \(\lambda_G\): \[ p_G=100\left(\lambda_G-1\right) \]

If \(p_G > 0\), LGSI efficiency is greater than LPSI efficiency; if \(p_G = 0\), both are equal, and if \(p_G < 0\), LPSI is more efficient. Additionally, accounting for the interval lengths between selection cycles, LGSI will be strictly more efficient than LPSI when: \[ L_G < \frac{\rho_{HI_G}}{h_I}L_P \]

### Genomic Estimated Breeding Values (GEBVs)

Because the `selection.index` package does not perform prediction modeling itself, the GEBVs must be calculated prior. In this example, we generate GEBVs (\(\boldsymbol{\gamma}\)) using an unpenalized Ridge Regression, where markers estimate trait values. For a rigorous methodology it is acceptable to generate GEBVs using Bayesian Alphabets, rrBLUP, or equivalent prediction models.

    library(selection.index)
    library(MASS) # For robust ridge regression implementation (lm.ridge)

    # 1. Load the built-in synthetic maize datasets
    data("maize_pheno")
    data("maize_geno")

    # Ensure markers are numeric matrices
    X <- as.matrix(maize_geno)

    # Ensure pheno lines up perfectly with geno
    # (Note: maize_pheno contains 4 repetitions per genotype; we take the means)
    Y_agg <- mean_performance(
      data = maize_pheno[, c("Yield", "PlantHeight", "DaysToMaturity")],
      genotypes = maize_pheno$Genotype,
      replications = maize_pheno$Block
    )
    #> Warning in sqrt(EMS/r): NaNs produced
    #> Warning in sqrt(2 * EMS/r): NaNs produced
    #> Warning in sqrt(2 * EMS/r): NaNs produced
    #> Warning in sqrt(EMS): NaNs produced
    #> Warning in sqrt(EMS/r): NaNs produced
    #> Warning in sqrt(2 * EMS/r): NaNs produced
    #> Warning in sqrt(2 * EMS/r): NaNs produced
    #> Warning in sqrt(EMS): NaNs produced
    #> Warning in sqrt(EMS/r): NaNs produced
    #> Warning in sqrt(2 * EMS/r): NaNs produced
    #> Warning in sqrt(2 * EMS/r): NaNs produced
    #> Warning in sqrt(EMS): NaNs produced

    # Sort both datasets to ensure identical ordering
    Y_agg <- Y_agg[order(Y_agg$Genotypes), ]
    X <- X[order(rownames(X)), ]

    # Ensure overlapping genotypes
    common_genotypes <- intersect(Y_agg$Genotypes, rownames(X))
    Y_agg <- Y_agg[Y_agg$Genotypes %in% common_genotypes, ]
    X <- X[rownames(X) %in% common_genotypes, ]

    # Select only multi-traits relevant to breeding
    # Yield, PlantHeight, DaysToMaturity
    Y <- as.matrix(Y_agg[, c("Yield", "PlantHeight", "DaysToMaturity")])

We now calculate \(\boldsymbol{\gamma}\) (the `gebv_mat`):

    # Simulate Genomic Estimated Breeding Values (GEBVs) using Ridge Regression
    # In best practice, you'd use cross-validation or a separate training/testing set
    # We use lambda = 100 to handle the p >> n dimensionality problem
    gebv_mat <- matrix(0, nrow = nrow(Y), ncol = ncol(Y))
    colnames(gebv_mat) <- colnames(Y)

    for (i in seq_len(ncol(Y))) {
      # Fit a ridge regression model for trait `i` using markers `X`
      model_ridge <- lm.ridge(Y[, i] ~ X, lambda = 100)

      # Predict values using coef() which correctly un-scales the coefficients: Include intercept
      betas <- coef(model_ridge)
      intercept <- betas[1]
      beta <- betas[-1] # Exclude intercept

      # Calculate the GEBV for the trait
      gebv_mat[, i] <- intercept + (X %*% beta)
    }

    head(gebv_mat, 3)
    #>         Yield PlantHeight DaysToMaturity
    #> [1,] 8241.151    219.4007       115.8738
    #> [2,] 7965.285    208.8754       116.4623
    #> [3,] 7832.597    204.5494       116.4592

### Implementing LGSI

With `gebv_mat` in hand, it is simple to calculate the LGSI metrics using `lgsi()`:

    # 2. Compute Trait Covariances
    pmat <- cov(Y) # Phenotypic Covariance (Approximation)

    # In optimal practice, the true Genomic Covariance Matrix (\Gamma) is
    # estimated using Restricted Maximum Likelihood (REML) utilizing both general
    # phenotypic and genotypic information. Here, we simulate \Gamma as proportional
    # to the phenotypic variance (assuming a high heritability correlation).
    gmat <- pmat * 0.4 # Approximate Genotypic Covariance (Approximating \Gamma)

    # 3. Define Economic Weights
    weights <- data.frame(
      Trait = c("Yield", "PlantHeight", "DaysToMaturity"),
      Weight = c(5, -0.1, -0.1)
    )
    wmat <- weight_mat(weights)

    # 4. Calculate Linear Genomic Selection Index (LGSI)
    # For the testing population where we only use genomic values
    lgsi_result <- lgsi(
      gebv_mat = gebv_mat,
      gmat = gmat,
      wmat = wmat
    )

    # Output Summary
    lgsi_result$summary
    #>   b.1  b.2  b.3       GA      PRE hI2 rHI
    #> 1   5 -0.1 -0.1 1718.838 171883.8   1   1

The LGSI function output shows the calculated standard deviation (\(\sigma_I\)), index heritability (\(h^2\)), the overall net genetic advance (\(GA\)), and expected gains per trait (\(\Delta H\)) associated with genomic selection.

## 5.2 The Combined Linear Genomic Selection Index

The Combined LGSI (CLGSI) developed by Dekkers (2007) is a slightly modified version of the LMSI, which, instead of using the raw marker scores, uses the GEBVs and the phenotypic information jointly to predict the net genetic merit.

The CLGSI model represents: \[ {I}_C={\boldsymbol{\beta}}_{\mathbf{y}}^{\prime}\mathbf{y}+{\boldsymbol{\beta}}_G^{\prime}\boldsymbol{\gamma} \]

Where \(\mathbf{y}\) is the predicted phenotype and \(\boldsymbol{\gamma}\) is the subset of genetic markers. As \(p \rightarrow \infty\), the true variance approaches the genomic predicted variance, and CLGSI becomes identical to LGSI. Because the CLGSI uses GEBVs and phenotypic information jointly, it can ideally complement prediction workflows inside training populations where field trials are simultaneously held.

### Statistical Properties of the CLGSI

Similar to LGSI, the CLGSI holds strict conditions ensuring theoretical validation constraints: 1. \(\sigma_{I_C}^2 = \sigma_{HI_C}\), meaning the variance of \(I_C\) and the covariance between \(H\) and \(I_C\) are unified. 2. The maximized correlation between \(H\) and \(I_C\) operates strictly as \(\rho_{HI_C} = \frac{\sigma_{I_C}}{\sigma_H}\). 3. The variance of the predicted error, \(Var(H - I_C) = (1 - \rho_{HI_C}^2)\sigma_H^2\), is mathematically minimal. 4. The total variance of \(H\) explained by \(I_C\) scales linearly via \(\sigma_{I_C}^2 = \rho_{HI_C}^2 \sigma_H^2\).

### Implementing CLGSI

You can measure the CLGSI utilizing the `clgsi()` function and feeding it both the observed phenotypes and estimated genomic variants.

    clgsi_result <- clgsi(
      phen_mat = Y, # Observed phenotypic data
      gebv_mat = gebv_mat, # Genomic Estimated Breeding Values
      pmat = pmat, # Expected Phenotypic traits covariance
      gmat = gmat, # Expected Genotypic traits covariance
      wmat = wmat
    )

    clgsi_result$summary
    #>       b_y.1    b_y.2    b_y.3    b_g.1    b_g.2    b_g.3       GA      PRE hI2
    #> 1 -241.2056 -69.0278 -11711.7 301.3417 -12.6428 13421.05 9325.477 932547.7   1
    #>   rHI
    #> 1   1

Notice that the Combined LGSI has two sets of optimized unitless \(\beta\) weightings—one for the phenotype values (`b_y`), and one for the overall marker value (`b_g`) per trait. Selection Response (\(R\)) represents the final expected genetic aggregate improvement.

## Limitations and Caveats

1. Dimensionality Problem: Often \(p \gg n\), requiring regularization techniques like Ridge Regression (as demonstrated in `lm.ridge`) to derive the Genomic Estimated Breeding Values without singular/non-invertible covariance anomalies.
2. Cycle Length: The genomic methods (LGSI, CLGSI) may occasionally show a lower index accuracy unadjusted than the phenotypic methods initially, but typically provide vastly more improvement per unit of time (\(L_G\)) because of how heavily compressed cycle iterations are when phenotyping is skipped.